Is it possible to get all available fossils on my island?

All islands have the same fossils. However random is as random does, and it's entirely possible you've had a slew of bad luck.

Keep at it! You'll get there eventually! I've a friend who's not traded fossils with anyone else, and he's only 3 fossils shy of completing his fossil set.

I don't know if every fossil is equally likely to appear, but even if it is you should expect to see quite a few duplicates before you pick up the last fossil for your collection.

Remember that when your collection is only halfway complete, already half of your identified fossils will be duplicates: at the three-quarter mark, only a quarter of identified fossils will be new, etc. etc.

(Mathematically speaking, if you're picking randomly from a set of N things, you'll need about N log N picks (natural log, base e) to get them all, so you should expect an average of log N duplicates of each thing before you get the last thing you need. In practice this will mean many duplicates of the things you picked up at the beginning and few or no duplicates of the ones you only got near the end. I didn't figure out the variance, but I did run some simulations, and eyeballing it it seems like it can vary significantly depending on luck.)